A337418 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 3) missing two edges, where the removed edges are not incident to the same vertex in the 3 point part but are incident to the same vertex in the other part.

32, 290, 2240, 16322, 116192, 819170, 5751680, 40314242, 282357152, 1976972450, 13840224320, 96885821762, 678213506912, 4747532812130, 33232844476160, 232630255706882, 1628412823069472, 11398892860850210, 79792259324043200, 558545843162577602

Offset: 3

Author

Steven Schlicker, Aug 26 2020

Keywords

• easy
• nonn

Comments

The Hausdorff metric defines a distance between sets. Using this distance we can define line segments with sets as endpoints. Create two sets from the vertices of the parts A and B (with |A| = 3) of a complete bipartite graph K(3,n) (with n at least 3) missing two edges, where the removed edges are not incident to the same vertex in A but are incident to the same vertex in B. Points in the sets A and B that correspond to vertices that are connected by edges are the same Euclidean distance apart. This sequence tells the number of sets at each location on the line segment between A and B.

Number of {0,1} 3 X n (with n at least 3) matrices with two fixed zero entries in the same column and no zero rows or columns.

Take a complete bipartite graph K(3,n) (with n at least 3) having parts A and B where |A| = 3. This sequence gives the number of edge covers of the graph obtained from this K(3,n) graph after removing two edges, where the removed edges are not incident to the same vertex in A but are incident to the same vertex in B.

Links

• Paolo Xausa, Table of n, a(n) for n = 3..1000
• Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
• Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Crossrefs

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

Programs

Maple

a:= proc(n) 7^(n-1)-2*3^(n-1)+1 end proc: seq(a(n), n=3..20);

Mathematica

A337418[n_] := 7^(n-1) - 2*3^(n-1) + 1;
Array[A337418,25,3] (* Paolo Xausa, Jul 22 2024 *)

PARI

Vec(2*x^3*(16 - 31*x + 21*x^2) / ((1 - x)*(1 - 3*x)*(1 - 7*x)) + O(x^25)) \\ Colin Barker, Nov 20 2020

Formula

a(n) = 7^(n-1)-2*3^(n-1)+1.

From Colin Barker, Nov 20 2020: (Start)

G.f.: 2*x^3*(16 - 31*x + 21*x^2) / ((1 - x)*(1 - 3*x)*(1 - 7*x)).

a(n) = 11*a(n-1) - 31*a(n-2) + 21*a(n-3) for n>5. (End)